This question actually has more to do with the intellectual history of the region than the power and importance of Greece itself. Ancient Greece, primarily Athens, is seen as the forerunner to the Romans, which in turn is seen as the forerunner to any number of western European ruler. In scholarship, this has manifested in a celebration of classical Greece and a devaluation of Hellenistic Greece (the period of and after Alexander) as scholars "wait" for Rome to come into its own a few centuries later. Recent scholarship is interested in putting ancient Greece into its own context -- Ancient Near Eastern rather than Mediterranean, necessarily, since Greece had way more to do with Asia Minor and Persia than northwestern Europe.

The corrective aspect of modern scholarship often runs into modern ideological/political issues -- the issues of Greek vs Macedonian cultural identity, for example, has been a recent issue, as you can see with this letter to US President Barack Obama about the "misuse" of Greek history by the modern republic of Macedon (http://macedonia-evidence.org/obama-letter.html).

Great question. I’m curious to see if others are going to chime in on this as it is a topic very open to discussion in my opinion.

You are right that their position in the Mediterranean was certainly one of the factors.

The seasonal climate with a lot of sun provided the necessary means for a sort of outdoor life. The Aegean sea provided the opportunities for honing skills in colonization, seafaring and larger scale trade. The Greeks had some very successful colonies in Magna Graecia (Great Greece) to the west that held great strategic importance later on. They were also close to an array of older and already successful societies and civilizations and could perhaps learn from them, like the Romans later adopted a lot of the Greek ideas and philosophy.

A few of these factors may have been present in other parts of the world (like the Baltic or southern Australia), but they certainly were not present all at once like with the Greeks.

The Persian Wars also had a say in creating a sort of ’us versus them’ mentality in the Greek societies, and the Greeks who did not fall under Persian dominance got more of a sense of identity, where previously the earlier Greeks did not think in this way, but instead quite the opposite. ’The West’ was now seen as glorious and having great wisdom, compared to the East.

Source: Norman Davies: ’Europe: A history’, Chapter II.

*them